Counting Reeb Chords on spherizations

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Counting Reeb Chords on

Spherizations

THÈSE DE DOCTORAT

présentée à la Faculté des Sciences de l’Université de Neuchâtel pour obtenir le grade de docteur ès sciences par

Raphael Elias Wullschleger

soutenue avec succès le 12 septembre 2014 et acceptée sur proposition du jury

Prof. Dr. Felix Schlenk Université de Neuchâtel, directeur de thèse Prof. Dr. Alain Valette Université de Neuchâtel

Prof. Dr. Urs Frauenfelder Universität Augsburg, Allemagne

Institut de Mathématiques Université de Neuchâtel

Rue Emile–Argand 11 CH–2000 Neuchâtel

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Faculté des sciences Secrétariat-décanat de Faculté Rue Emile-Argand 11 2000 Neuchâtel - Suisse Tél: + 41 (0)32 718 2100 E-mail: [email protected]

IMPRIMATUR POUR THESE DE DOCTORAT

La Faculté des sciences de l'Université de Neuchâtel

autorise l'impression de la présente thèse soutenue par

Monsieur Raphael WULLSCHLEGER

Titre:

“Counting Reeb Chords on Spherizations”

sur le rapport des membres du jury composé comme suit:

- Prof. Felix Schlenk, Université de Neuchâtel, directeur de thèse - Prof. Alain Valette, Université de Neuchâtel

- Prof. Urs Frauenfelder, Universität Augsburg, Allamagne

Neuchâtel, le 9 octobre 2014 Le Doyen, Prof. B. Colbois

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Summary

In classical physics, one is interested in finding solutions of the Newtonian equations of motion. If there is a certain number of bodies which attract each others and if one assumes an initial configuration of these masses, then one would like to understand the time evolution of this system according to Newton’s equations, i.e. the change of position and momentum of all these bodies as functions of time. But already in the case of three bodies – say, the moon, the sun and the earth – one knows only very little and this question remains essentially unanswered.

Rewriting the Newtonian equations of motion in an equivalent way leads to Hamil-ton’s equations. Solutions of HamilHamil-ton’s equations are paths – in physical terms – in phase space, whereas in mathematical terms one calls this space the cotangent bundle. So classical physical evolution takes mathematically place in cotangent bundles.

Symplectic geometry is a new and prominent subject within differential geometry, one of the few basic branches of mathematics. The cotangent bundle is probably the most famous representative of a so-called symplectic manifold. It holds true that the old physical questions got via the steps explained above a new and strong geometrical interpretation.

Floer homology is a powerful tool to study solutions of Hamilton’s equations. It gives the possibility to use topological information about the cotangent bundle to obtain qualitative and quantitative results on solutions of Hamilton’s equations.

The energy is a property of a physical system which remains constant during evo-lution of time. Therefore, it is natural to look for soevo-lutions of Hamiltonian systems on surfaces as certain subsets – called energy hypersurfaces – of cotangent bundles which are characterized by the fact that the energy function takes for all points of these sur-faces the same value. Roughly speaking, solutions of Hamilton’s equations along energy

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hypersurfaces are called Reeb chords. The spectrum of such an energy hypersurface is simply the set of all times needed to move along the paths which are solutions of Hamilton’s equations. So it is the set of times required to walk along the Reeb chords of a given energy hypersurface. In particular, the counting function associated to an energy hypersurface is studied. This function calculates the number of solutions whose times are shorter than a given value.

In this thesis, steps are taken towards an understanding of the time spectrum of fiberwise starshaped hypersurfaces in cotangent bundles. The base manifold is through-out assumed to be a closed connected Riemannian manifold. It is shown that under the additional assumption of exponential- resp. polynomial growth of the fundamental group of the base manifold, the counting function grows at least exponentially resp. at least polynomially in time. Generally, for every fiberwise starshaped hypersurface over a closed connected Riemannian manifold, the associated counting function grows at least linearly in time. These are asymptotic results. Afterwards the question of understand-ing fast Reeb chords is considered. An estimate for the time of the fastest resp. of the second fastest Reeb chord is given. More specifically, this question is addressed by choosing special base manifolds, or configuration spaces, such as Lie groups or generally (Riemannian) symmetric spaces. Estimates for the times of the k fastest Reeb chords are deduced. These estimates depend on the geometry of the base manifold only. Another attempt is of group theoretic nature. If the fundamental group of the base manifold is of order k, then there are at least k Reeb chords satisfying an upper time bound k times the diameter of the (compact) base manifold. Finally, some results concerning the stability of the time of the fastest Reeb chord are presented.

Raphael Wullschleger

Keywords. Hamiltonian Dynamics; Symplectic Geometry; Lagrangian Floer Homol-ogy; Contact Geometry; Reeb Dynamics

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Acknowledgements

First of all, I would like to express my gratitude to my supervisor Professor Dr. Felix Schlenk. He offered me the position as a PhD student in a fascinating and modern field of mathematics. This was the starting point of truly interesting, challenging and beautiful four years of research and teaching sessions at the University of Neuchâtel. During this time I learnt a lot from him. His deep insights which led to the good and natural questions, his enthusiasm for mathematics impressed me. Felix’ continuous interest in my questions, his careful consideration of my ideas, accompanied by his patience and generosity was a great help and a true support on my way. I wish Felix for the coming years a lot of time for his family, for challenging hiking tours, time to pursue his broad interests in literature, culture and certainly further subjects.

Many thanks go to Professor Dr. Urs Frauenfelder from the University of Augsburg in Germany and to Professor Dr. Alain Valette from the University of Neuchâtel. Very kindly, they accepted to be referees for my defence and provided enlightening hints and advice which allow further steps.

I would like to thank Dr. Alexandre Girouard and Dr. Antoine Gournay. When I started my doctoral studies in Neuchâtel, Alexandre and Antoine were the two Maître as-sistants in the institute. They helped me a lot with their broad mathematical knowledge, with ingenious ideas, with a series of good suggestions and sometimes with organisa-tional steps which made my life easier. Alexandre moved to Montréal in Canada, I wish him all the best for his future. This autumn, Antoine left Neuchâtel in the direction of Dresden, Germany. I send him and his young family my best wishes and a happy time. Thanks to the invitation of Dr. Yaron Ostrover from the University of Tel Aviv, I had the pleasure to spend a couple of days in Israel to pursue research on “Hofer-Zehnder capacity and convex billiards”. Many thanks to Yaron for giving me this opportunity.

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Then, I also would like to express my gratitude to the European Science Foundation (ESF) who funded in a fast and non-bureaucratic way my stay in Tel Aviv via the ESF activity “Contact and Symplectic Topology” (CAST).

Many thanks go to Elena Kantonistova. I met her on the occasion of a summer school in Cologne, Germany. She offered me very kindly to translate a Russian article by Albert Schwarz into the french language. I am sure that Elena needed a lot of patience and energy to answer to all my questions interrupting the translation process. Thanks to her work, I got a better understanding of some important aspects of this thesis. I wish Elena all the best and a successful time at the Lomonossow University in Moscow.

A grand “merci bien” goes to all the students with whom I met during my time in Neuchâtel. I think, we shared a lot of great and hilarious moments during the exercise classes and outside the class rooms, I highly liked to be an assistant. In particular, I would like to say thanks for their confidence which they showed to me as their assistant and for their patience when I struggled with the french language.

“Un grand merci” goes to Muriel Heistercamp and to Régis Straubhaar, then “ein grosses Dankeschön” is sent to Cologne in Germany, to Dorothee and her husband Thanasis Stylianou. Usually, due to their initiative, we had a series of great and unfor-gettable moments. With Alice Badin I shared a lot of great moments during my time in Neuchâtel. Thanks to her generous and social attitude, many things went well, I recall many great lunches in the cafeteria and conversations about this and that in our offices. Many thanks also to Alberto Ravagnani for all the interesting conversations and all the hilarious moments. It was “molto bene” – his incredibly good Pizza for example.

I would like to say thanks to all my colleagues at the institute for all the interesting conversations, the nice invitations, good apéros and great events which were organized yielding a very good atmosphere.

Finally, I would like to express my deep gratitude to my family for the broad, strong and ongoing support over all the years. Their believe in my goals was important.

I am deeply grateful to Panna for all her help and understanding during the last months of intensive work, for her great generosity and her endless patience. Köszönöm szépen, szerelmem.

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Meinen Eltern gewidmet

Verena Wullschleger–Basler

und

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Notations and Symbols

N := {1, 2, 3, ...} The natural numbers

N0 := {0, 1, 2, 3, ...} The natural numbers with 0

Z := {..., −2, −1, 0, 1, 2, ...} The integers

R+ := {x ∈ R | x > 0} The strictly positive real numbers

R+0 := {x ∈ R | x ≥ 0} The non-negative real numbers

Fp A series of coefficient fields, see Section 2.2.3

Z2 := F2 = Z/2Z The set of integers Z modulo 2

M, (M, g) A closed (i.e. compact without boundary) connected finite-dimensional smooth mani-fold. Usually furnished with a Riemannian metric g, we speak of the Riemannian mani-fold (M, g)

q, q0 A pair of points in M . The point q is the starting point of a geodesic segment, whereas q0 is the point where this curve ends

d := diam M The diameter of the manifold (M, g)

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T_x∗M The fiber of the cotangent bundle T∗M over the base point x ∈ M

Σ A fiberwise starshaped hypersurface in T∗M

Σx For x ∈ M : Σx := Σ∩Tx∗M , see Section 2.1.2

H A function H : T∗M → R, called the Hamil-tonian, see Section 2.1.1

AH, A , T The Hamiltonian action functional

associ-ated with H and the reduced action or the time, see Section 2.1.1

P(H, q, q0_{) , P}b_{(H, q, q}0₎

The set of solutions of Hamilton’s equations starting in T_q∗M and ending in T_q∗0M ; b

indi-cates an upper action-bound

G−, K, G+ Three special Hamiltonians, see Section 2.2

U (q, H), V (q) q ∈ M and H : T∗_{M → R a Hamiltonian} function, see Definitions (2.11), (2.12) CFq,q0_,Σ The counting function counts Reeb chords

from Σq to Σq0, defined in (2.9)

CFk Floer chain groups of Conley–Zehnder

in-dex k

Cp Cp := {q ∈ M | x conjugate to q} is the set

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Abstract Mathematics is about “interesting structures”. What makes a structure interesting is

an abundance of interesting problems; we study a structure

by solving these problems.

M. Gromov, [20]

Chapter 1 Introduction

The first chapter shall describe in detail the questions addressed in this thesis and the results obtained. Starting from basic physical principles, we will focus on the naturality and the importance of the problems considered, and we will place the topic in the broader context of mathematical research. By doing so, we follow the books of Arnol0d [7] and of Hofer–Zehnder [25]. Finally, we give an overview of the results obtained.

1.1 The questions of this thesis

The roots of the questions of this thesis lie in physics. Classical mechanics is the first analytic approach to describe physical phenomena. This theory focuses mainly on un-derstanding the time evolution of the positions of physical bodies which exert forces on each others. The definitions and principles were introduced in the Mathematical Principles of Natural Philosophy (Philosophiæ Naturalis Principia Mathematica) by I. Newton in the year 1687.

Let us consider the motion of a certain number n of point-mass particles in three-dimensional real space R3_{. The totality of these n particles forms the physical system.}

More precisely, if we fix one of these n particles, say Pi, where i ∈ {1, ..., n} specifies our

choice, the time evolution of the position of Pi can be described by a coordinate map,

xi : R −→ R3,

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The variable t refers to time. These coordinate maps shall assumed to be at least twice continuously differentiable mappings. As an illustration we can consider the so-called world lines traced by these points under time evolution, see Figure 1.1.

t0 time t x1 x2 x1(t0) x2(t0) R3 ⊂ R3 × R

Figure 1.1: This figure shows the world lines (as paths in “space–time” R3 _{× R) of}

two particles P1 and P2. The horizontal axis stands for the time parameter of the

two coordinate maps. Observe that, possibly, there is a (collision) time t0 for which x1(t0) = x2(t0) holds true and in addition, that the vector x(t) := (x1(t), x2(t)) ∈ R6 is

a point in six-dimensional real space.

So far, we treated the n particles, or the n bodies, separately. In physics, one is interested in interactions between the particles or in understanding the forces exerted on a given body by the other bodies. Therefore, it is natural to define a mapping depending on time and describing the positions of these n particles at once,

x : R −→ R3n_,

t 7−→ x(t) := (x1(t), . . . , xn(t)) ,

where R3n _{stands for the direct product of n copies of R}3_{, just because all the n particles}

can move freely in three-dimensional space. Since the xi are differentiable mappings, we

can consider their derivatives

˙xi(t0) = dxi dt t=t0 ∈ R3,

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called the velocity vectors at the time t0, as well as the acceleration vectors at t0 ¨ xi(t0) = d2xi dt2 t=t0 ∈ R3_.

Newton’s principle of determinacy, one of the principles of classical mechanics, im-plies that the initial positions (x0)i of all n point-masses and their initial velocities

(v0)i uniquely determine the motion of the system. So, the vector of initial positions

x0 = ((x0)1, ..., (x0)n) and the initial velocities v0 = ((v0)1, ..., (v0)n) must determine also

the acceleration of any body at any time. Mathematically, this means that there exists a map F (F for force)

F : Ω ⊂ R3n× R3n

× R −→ R3n

such that

¨

x(t) = F (x(t), ˙x(t), t) . (1.1) The equations (1.1) are the well-known Newtonian equations of motion of classical me-chanics. Observe that we set all masses mi equal to one. The map F introduced above

is found by experimental means. Forces are measurable. Note that the domain Ω of F is often a strict subset of R3n× R3n_{× R. This is the case for example if one considers}

two bodies and the forces exerted on them due to gravity. One has to exclude the points of collision, since for these F is not defined.

Newtons equations of motion (1.1) form a system of ordinary differential equations for the time evolution of the positions xi, or the trajectories, of the n point-masses. As

explained at the beginning, we are interested in finding these trajectories to predict the time evolution of the considered physical system.

A physical system is said to be conservative if there exists a continuously differen-tiable function U

U : Ω ⊂ R3n−→ R such that

F (x(t), ˙x(t), t) = −∇U (x(t)). (1.2) This function U is called the potential or the potential function. As it is apparent from

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equation (1.2), the function U evaluated at a point is formally an energy and describes the energy of a particle according to its position relative to the others. For such a system of n particles we introduce its total kinetic energy (which is the sum of the individual kinetic energies of the n particles):

K( ˙x(t)) := 1 2 n X i=1 ˙xi(t)2.

We assumed that all the n bodies are point-mass particles, so they do not have any spatial extent what could lead to rotational energy or similar. Then we can speak of the total energy of the considered system:

E(x(t), ˙x(t)) := K( ˙x(t)) + U (x(t)). (1.3)

The following theorem points out the very important property of conservative systems. For a proof, see Theorem 1.1.4.

Theorem 1.1.1 (Energy conservation). Let x be a solution of equation (1.2), then the total energy E is constant along this solution x, i.e.

∀ t0 ∈ R : dE dt (x(t), ˙x(t)) t=t0 = 0.

So far, we derived Newton’s equations of motion and we tried to point out why we are interested in finding their solutions. The next step is to show that the solutions of a conservative system can be determined via a variational principle, “Hamilton’s principle of least action”. The calculus of variations is concerned with the extremals of functions, or functionals, whose domain is an infinite-dimensional space, the space of all curves from one point to another one.

Let us consider the Lagrangian or the Lagrange function of a conservative system, defined by

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and the associated action functional

Φ(γ) = Z t1

t0

L(γ(t), ˙γ(t), t) dt , (1.5)

where γ : [t0, t1] → R3n is a C2-path in R3n. A theorem says that the extremals, formally

iven by dΦ(γ) = 0, see Arnold [7] for details, coincide with the solutions of Newton’s equations of motions,

¨

x = −∇U (x).

This is Hamilton’s principle of least action. Suppose that γ : [t0, t1] → R3n is an

extremal of the functional Φ. Then it is necessary and sufficient that γ satisfies the so-called Euler-Lagrange equations

d

dt∇x˙L(γ(t), ˙γ(t)) − ∇xL(γ(t), ˙γ(t)) = 0. (1.6) This variational principle opens the door to the vast area of the calculus of variations. But the drawback from the point of view of mathematics is still there: the Euler-Langrange equations (1.6) are in fact a system of n second-order ordinary differential equations.

The Euler-Lagrange equations (1.6) are evaluated at points lying on paths x defined on a time interval having values in the space R3n_{, for example x : [0, 1] → R}3n_{. In}

particular, this holds true for very short segments of this path x. These segments can be viewed as short segments of paths in a chart of some manifold M . Therefore, we can consider our situation in the more general setting of manifolds. This means formally that the Lagrangian is a smooth function on the tangent bundle T M of M ,

L : T M _{−→ R ,}

(x, vx) 7−→ L (x, vx) .

(1.7)

Note that we omit here an explicit time-dependence of L – see below for an explanation – and also that we allow all possible dimensions dim M = n for the manifold M . Assumed constraints on the physical space reduce the dimension of M by one, two or higher. Denote by T∗M the cotangent bundle of M and define Lx(·) := L(x, ·) : TxM → R.

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Definition 1.1.2 (Fiberwise Legendre transform map). Let M be a smooth manifold and L a Lagrangian. The fiberwise Legendre transform map of L, or Legendre transform map of Lx, is the map

L : T M −→ T∗_{M ,}

(x, vx) 7−→ L (x, vx) := (x, dLx(vx)) .

Note that the differential dLx(·) : TxM → Tx∗M evaluated at the point vxgives dLx(vx) ∈

T_x∗M . Hence, (x, dLx(vx)) ∈ T∗M . We define the fiberwise Legendre transform of L by

H : T∗M _{−→ R ,}

(x, ax) 7−→ H(x, ax) := supv∈TxM(ax(v) − L(x, v)) .

This function H is usually called the Hamiltonian or the Hamilton function associated to L. Note that if L is of the form K − U , then H is of the form K + U . So in this case H coincides with the total energy E of the system.

Let U ⊂ T M be an open set, choose coordinate functions qi, wi, on U, i ∈ {1, ..., n}.

If L is fiberwise strictly convex, i.e. if the fiberwise Hessian of L is positive definite

det ∂2L ∂wi∂wj > 0 ,

then the fiberwise Legendre transform map L of L is a diffeomorphism. For a given path γ : R → M , consider its analog in the cotangent bundle, given by the Legendre transform map

L(γ)(t) := (γ(t), pγ(t)) := L (γ(t), ˙γ(t)) .

Theorem 1.1.3. The path γ : R → M is a solution of the Euler-Lagrange equations (1.6) if and only if the path L(γ) : R → T∗M satisfies the so-called Hamilton equations,

d dtpi = − ∂H ∂qi , d dtqi = ∂H ∂pi ,

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One can write down the Hamilton equations in the following compact form ˙z(t) = J ∇H(z(t)) , where J :=   0 − Idn Idn 0  .

Hamilton’s equations build a system of 2n first-order differential equations. This is in contrast to the system of n second-order differential equations given by the Euler-Lagrange equations, (1.6). We will make the abbreviation XH := J ∇H and call it

the Hamiltonian vector field, see below for the details. An energy hypersurface of the cotangent bundle is a regular level set of the Hamiltonian H : T∗_{M → R. This energy} hypersurface will be denoted by Σ. In this thesis, we will study the solutions of the dynamical system

˙γ(t) = XH(γ(t)) , γ : R → T∗M

on energy hypersurfaces. Solutions of Hamilton’s equations must lie on such energy hypersurfaces, see Theorem 1.1.4 below.

Let us describe the setting of our questions. Consider the cotangent bundle T∗M of a n-dimensional closed connected Riemannian manifold (M, g), where g is the Riemannian metric on M . A symplectic structure on a smooth manifold P is a non-degenerate closed 2-form ω ∈ Ω2_{(P ), see Definition 2.1.1. Choose a point x = (x, ξ}

x) ∈ T∗M and consider

the standard projection π : T∗M → M defined by (x, ξx) 7→ x. We then are able to

define globally a differential 1-form,

λ(x)(vx) := ξx(dπ(x)(vx)) , vx ∈ Tx(T∗M ) . (1.8)

This is the so-called Liouville or tautological 1-form, see Hofer–Zehnder [25]. The cotangent bundle T∗M equipped with the 1-form (1.8) leads to a symplectic mani-fold (T∗M, dλ). Choose a hypersurface Σ in T∗M which is fiberwise starshaped with respect to the origin, i.e. Σx := Σ ∩ Tx∗M is strictly starshaped with respect to the zero

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everywhere. An example is the 1-form λ|Σ. A contact form η determines a contact

structure on Σ, the oriented hyperplane field ξ := ker(η) ⊂ T Σ. In our case this shall be ξΣ := ker(λ|Σ). Choose a smooth function H : T∗M → R which is fiberwise

ho-mogeneous of degree two such that Σ is a regular level set of H, so Σ is an energy hypersurface. The Hamiltonian vector field XH belonging to the function H is defined

by dλ(XH, ·) = dH(·). Since all levels are compact, this vector field has a flow ϕt

satisfying Hamilton’s equations d dtϕ

t_{(x) = X}

H ϕt(x) , x ∈ T∗M . (1.9)

We explained above that the solutions of the Euler-Lagrange equations (1.6) coincide with the extremals of the action functional Φ (1.5). A main object of this thesis is the Hamiltonian action functional which generalizes (1.5) to the setting of cotangent bundles: Let Ω1_q,q0T∗M be the space of paths of W1,2-Sobolev type in T∗M on the unit interval

[0, 1] ⊂ R from the point γ(0) ∈ Σq to the point γ(1) ∈ Σq0. Then we can introduce the

Hamiltonian action functional

AH : Ω1q,q0T∗M −→ R , γ 7−→ AH(γ) := R γλ − R1 0 H(γ(t)) dt .

In accordance with what we said concerning the Lagrangian action functional, it is well-known that the solutions of (1.9) are the critical points of the Hamiltonian action functional AH. In the field of symplectic geometry, Floer homology is a powerful tool to

study the critical points of the Hamiltonian action functional. Floer homology is a Morse theory for this functional. So it provides the possibility to use topological information about the cotangent bundle T∗M to get qualitative and quantitative results concerning the solutions of Hamilton’s equations (1.9). We will pursue this approach to get answers to our questions.

The following result generalizes Theorem 1.1.1.

Theorem 1.1.4 (Flow invariance of H). If ϕt_{is the flow of the Hamiltonian vector field}

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defined) H ϕt(x) = H(x) . Proof. Indeed, d dtH ϕ t_(x) = dH ϕt(x) · d dtϕ t_{(x) = dH ϕ}t_{(x) · X} H ϕt(x) = dλ XH ϕt(x) , XH ϕt(x) = 0 , by antisymmetry of dλ.

Recall that we assumed that the Lagrangian L (1.7) does not depend explicitly on the time t. This implies via the Legendre transform the same for the Hamiltonian H. If H depends explicitly on time, then this invariance property does not hold.

The contact form λ|Σ determines by the following two conditions the unique Reeb

vector field R on T Σ by

d(λ|Σ)(R, ·) ≡ 0 , λ|Σ(R) ≡ 1 .

The associated flow is called the Reeb flow ϕR of R. One can show that the Reeb flow

ϕR of ker(λ|Σ) is a reparametrization of the flow ϕH|Σ of XH restricted to Σ, we refer

to Lemma 4.8.4. A Reeb chord is a flow line of ϕR.

Question A. Is it true that for any two points q, q0 ∈ M , there exists a Reeb chord from Σq to Σq0 ?

This is a version of the Arnol0d Chord Conjecture for fibers of a starshaped hypersurface in the cotangent bundle. Let (C, ξ) be a (2 ` + 1)-dimensional contact manifold and let L be an integral submanifold of ξ. If dim L = ` then the submanifold L is called Legendrian, see [31]. The Arnol0d Chord Conjecture stated in [6] asks in the case of a contact manifold C for a Reeb chord which starts and ends in a given Legendrian submanifold L of C.

Consider the length spectrum of the Riemannian manifold (M, g) given by

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In this definition, the length is induced by the Riemannian metric g. The study of the length spectrum of a Riemannian manifold is an important problem, with many results and many open questions, see Berger [11, Chapter 10] and Paternain [42, Chapter 5].

In this thesis we are interested in a more general problem: We study the spectrum of fiberwise starshaped hypersurfaces Σ of the cotangent bundle T∗M . Define the set

S (Σ, q, q0) := {T (γ) | γ a Reeb chord on Σ from Σq to Σq0} ,

where the number T (γ) is the time needed by the Reeb chord γ to go from Σq to Σq0.

It is given by

T (γ) = Z

γ

λ ,

and where the numbers T (γ) are listed with multiplicities. Knowing the set σq,q0(Σ) is

equivalent to knowint the counting function

CFq,q0_,Σ(T ) := # {τ ∈ S (Σ, q, q0) | τ ≤ T } . (1.10)

If one fixes a time T , then the counting function gives the number CFq,q0_,Σ(T ) of Reeb

chords starting in the fiber Σq and ending in some point of Σq0 before or at the time T .

Question B. Is it possible to find a function fΣ : R → R with fΣ(T ) → +∞ ( T → +∞)

such that

CFq,q0_,Σ(T ) ≥ f_Σ(T ) > 0

independently of the points q, q0 ∈ M ?

1.2 Summary of the results

Let (M, g) be an n-dimensional closed connected Riemannian manifold and denote by d := diam M the diameter of M . Consider a fiberwise starshaped hypersurface Σ ⊂ T∗M in the cotangent bundle T∗M and a Hamiltonian function K : T∗_{M → R homogeneous} of degree two. Assume that 1₂ is a regular value of K such that Σ = K−1 1₂. We refer to Section 2.2 for details.

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In this section, we state the main results of the thesis and explain the connections between them. The main tools for analyzing the spectrum of spherizations are Morse theory, Floer homology, and results on the growth of finitely generated groups and of the homology of based loop spaces. Morse theory for the energy functional gives lower bounds for the number of geodesic paths between two non-conjugate points in terms of the homology of the based loop space of M .

Assume first that our Reeb flow is a geodesic flow on the Riemannian manifold (M, g). Assume that q, q0 are non-conjugate. We would like to understand the function CFq,q0_,Σ(t)

counting geodesics from q to q0 of length ≤ t. Since we are looking for lower bounds of this number that are “true for all metrics g”, we only look for homologically visible geodesics (Definition 4.1.18), and hence use Morse theory: Consider the energy func-tional Eg(γ) := 1 2 Z 1 0 g( ˙γ(t), ˙γ(t)) dt (1.11) on the space of candidates Ω1_q,q0M of W1,2-paths γ : [0, 1] → M with γ(0) = q and

γ(1) = 1. The critical points of Eg are precisely the geodesics from q to q0. If we

denote by E_ga(q, q0) the sublevel set {γ ∈ Ω1_q,q0M | Eg(γ) ≤ a}, and notice that for

a geodesic, twice the energy equals the length squared (Lemma C.3.4), the classical Morse-inequalities tell us that

CFq,q0_,Σ(t) ≥ dim H_∗ Et2/2 g (q, q 0 ), F = ∞ X j=0 dim Hj Et2/2 g (q, q 0 ), F (1.12)

provided that q, q0 are non-conjugate. Indeed, this condition is equivalent to saying that Eg is Morse. On the right hand side, H∗ denotes singular homology, and coefficients are

taken in a field F. The sum on the right hand side is finite, since Egt2/2(q, q0) is homotopy

equivalent to a finite dimensional CW-complex, see Milnor [32]. The inequality (1.12) looks wonderful, since it seems to translate our geometric-dynamical problem into a topological one. However, two questions arise:

Question 1. How can we understand dim H∗ E_ga(q, q0), F ?

Question 2. And what if q, q0 are conjugate ?

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estimate in (1.12), one should take the supremum over all fields F. By the universal coefficient theorem, it suffices to consider only one field per characteristic, say Q and the finite fields Fp for p prime. Note that in Chapter 4 we will prove related results for

special choices of F. For example we will choose F = Z2 in Section 4.2.3.

The numbers dim H∗ Ega(q, q 0

), F are in general too hard to compute. One reason is that they may depend rather irregularly on a, g and q, q0. In particular, the function a 7→ dim H∗ Ega(q, q

0

), F may not be monotone increasing. To remedy for these irregularities, we consider the numbers dim ιa

∗H∗ E_ga(q, q0), F instead. Here ιa∗: Ega(q, q

0_{) → Ω}1

q,q0M is

the inclusion. The number dim ιa∗H∗ Ega(q, q 0

), F is the dimension of the part of the homology of Ωq,q0M that can be represented by cycles in Ea

g(q, q

0_{). A cycle in E}a g(q, q

0_{) is}

still a cycle in Ω1_q,q0M , while it may be bounded in Ω1_q,q0M but not in E_ga(q, q0). Hence

dim H∗ E_ga(q, q0), F

≥ dim ιa

∗H∗ E_ga(q, q0), F .

The functions bg,q,q0(a) := dim ιa_∗H_∗ Ea g(q, q

0

), F are much better behaved: Clearly, they are monotone increasing in a. A more fundamental reason that we are interested in geodesic chords that are homologically visible in the total path space Ω1

q,q0M , and not

just in the sublevel E_ga(q, q0), is the following: We shall find Reeb chords by sandwiching the sublevel of Σ between two sublevels of Eg, and this will lead to a lower bound of

CFq,q0_,Σ(t) in terms of b_g,q,q0(a), but not in terms of dim H_∗ E_ga(q, q0), F. The following

theorem will provide lower bounds of CFq,q0_,Σ in terms of the homology H_∗ Ω1,a

q,q0M, F.

It is proven in Section 2.2.3. See the different parts of Section 2.2 for the definitions and notions used in the statement.

Proposition 1.2.1. Fix two points q, q0 ∈ M such that q0 _{is not Σ-conjugate to q. Let}

g be a Riemannian metric on M such that q, q0 are not g-conjugate. Scale g such that G ≤ F . Let a < b and σ ≥ σg ≥ 1 be such that

a, b, a/σ, σb /∈ S (G, q, q0) and a/σ, b /∈ S (F, q, q0) .

Then for any field F the number of Reeb chords from Σq to Σq0 in class α with action

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inclusion

e

H∗ QE(a/σ,b](q, q0, α) ; F → He∗ QE(a,σb](q, q0, α) ; F .

By eH∗ we denote reduced homology. The Proposition 1.2.1 is a consequence of the

Abbondandolo–Schwarz isomorphism, see [3], from the Floer homology groups of T∗M to the homology groups of the based loop space Ω1_q,q0M .

1.2.1 Exponential and polynomial growth of the number of

so-lutions

We give an overview on asymptotic results of the counting function CFq,q0_,Σ. By doing

so, we answer Question B in special cases.

Since M is a closed manifold, its fundamental group π1(M, q), for q ∈ M , is a finitely

presented group. Choose a finite set S of generators of π1(M, q). For each positive

integer m the function γS(m) counts the number of distinct elements in π1(M, q) which

can be written as words with at most m letters from S ∪ S−1. If the following limit is strictly positive, we say that π1(M, q) has exponential growth,

lim

m→+∞

log γS(m)

m ∈ [0, +∞) . (1.13)

Note that this limit exists, but depends on S.

Similarly, the polynomial growth of π1(M, q) is defined by

γ(G) := lim sup

m→+∞

log γS(m)

log m ∈ [0, +∞] . (1.14) Note that γ(G) does not depend on S.

See Section 2.5 for details and examples of spaces with fundamental groups of expo-nential resp. polynomial growth.

Theorem 1.2.2 (Exponential and polynomial growth of the number of solutions). Let q ∈ M . If π1(M, q) has exponential growth, then for every q0 ∈ M the number of orbits

of the flow ϕt_K|_Σ from Σq to Σq0 grows at least exponentially in time,

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Analogously, if π1(M, q) has polynomial growth k, it follows that for every q0 ∈ M the

counting function CFq,q0_,Σ grows at least polynomially in time,

CFq,q0_,Σ(t) < tk.

To give a partial answer to Question A and to Question B, let the fundamental group of M be finite.

Theorem 1.2.3 (Linear growth of Reeb chords). If π1(M ) is finite, then it follows that

CFq,q0_,Σ(t) < t .

The proof uses the well-known result of Serre [46] that for any simply-connected man-ifold M there is a sequence of integers (k`)_`∈N for which the Betti numbers bk`(ΩM, F)

of the corresponding based loop space ΩM are not zero.

1.2.2 Time bounds for the first and the second Reeb chord

In Section 1.2.1 about exponential and polynomial growth we gave asymptotic results on the counting function CFq,q0_,Σ. In particular, the constants appearing in the expressions

for the lower bounds are not well-understood. This issue shall be addressed next. Convention 1.2.4. Let (M, g) be furnished with a Riemannian metric g such that the Hamiltonian functions F, G defined in Section 2.2.1 satisfy

F ≥ G .

Denote by d := diam(M, g) the diameter of (M, g).

Chapter 5 covers the details on how to deduce a concrete upper bound on the times of the first two shortest Reeb chords from Σq to Σq0.

Due to basic geometric facts, we know that for every Riemannian metric g satisfying the Convention 1.2.4 the following estimate for the first Reeb chord holds. Denote by

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dist(q, q0) the distance of q and q0 with respect to g. Then,

T1(Σ, q, q0) ≤ dist(q, q0) .

Given a simply-connected manifold M , let k0 be the smallest integer k such that

Hk(M ; F) 6= 0. Then k0 ∈ {2, . . . , dim M }.

Theorem 1.2.5. Let Σ be a fiberwise starshaped hypersurface in T∗M with M simply-connected. Assume that g satisfies the Convention 1.2.4. If q0 is not Σ-conjugate to q, then T2(Σ, q, q0) ≤ 8k20d + (2k 2 0 − 1)3d √ σg − 1 < 2k2₀d 4 + 3√σg .

Recall that n is the dimension of M and d refers to the diameter d = diam(M, g). The existence of the short Reeb chords with the given time bound relies on results of Nabutovsky–Rotman, see [34] and for more details the sections in Chapter 5.

1.2.3 Time bounds for the first k Reeb chords – via Morse theory

Fix k ∈ N. In Chapter 4 we explane two approaches to get concrete upper bounds on the times of the first k Reeb chords on Σ.

We start with Morse theory in infinite dimensions under the assumption that the energy functional (1.11) is a so-called perfect Morse function on the space of candi-dates Ω1_q,q0M (for almost all pairs q, q0) with respect to F. The notion of a perfect Morse

function is explained in Section 6.1. This allows us to interprete k geodesic segments (on M ) with given length bounds as homologically visible, see the Definition 4.1.18. Their existence is guaranteed by the work of Nabutovsky–Rotman [37]. Via Proposition 1.2.1 we then get

Proposition 1.2.6. Let M be n-dimensional [2-dimensional]. If for q ∈ M and almost every q0 the energy functional Eg is an F-perfect Morse function on Ω1q,q0M , then for

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from Σq to Σq0 satisfying the time bound

T (˜x`) ≤ 2n(k + 1)2d ,

[ T (˜x`) ≤ (22k − 21)d ] .

The expression in square brackets accounts for the 2-dimensional case. In general it is a very hard problem to understand whether Eg is a perfect Morse function. This

question leads deeply into the field of algebraic topology.

Work done by Bott-Samelson [14] yields that Eg is perfect with respect to Z2

-coefficients on Ω1

q,q0M (for almost all pairs q, q0) if M is a compact Riemannian symmetric

space. This fact and Proposition 1.2.6 imply the following result:

Let (G, gbi) be an n-dimensional compact connected Lie group carrying a bi-invariant

Riemannian metric gbi and let H ⊂ G be a closed connected subgroup. Note that we

can scale gbi such that this metric satisfies the Convention 1.2.4 and is still bi-invariant.

Theorem 1.2.7. Let G/H be a compact symmetric space with the induced Riemannian metric. Further, let q, q0 _{∈ G/H be two arbitrary points and fix k ∈ N. Then there exist} k Reeb chords ˜x` on T∗(G/H) from Σq to Σq0 satisfying the time bound

T (˜x`) ≤ q 2 2 (n(k + 1)2_d)2_{+ d ,} " T (˜x`) ≤ s 2 (22(k − 1)d + dist(q, q 0₎₎2 2 + d # .

An analogous statement holds true if M is a compact simply-connected Lie group, see Proposition 4.2.20, but with respect to any coefficient field F. Therefore, we treat this special case individually.

Moreover, Chapter 4 consists of other results concerning the Conley–Zehnder index of the k Reeb chords ˜x` and manifolds M of non-positive curvature.

1.2.4 Time bounds for the first k Reeb chords – via topology

For the next step we pursue a group-theoretic approach: We forgo the last geometric restrictions on M and say something about quantitative existence results of Reeb chords

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on Σ under topological assumptions on the order of the fundamental group π1(M, q).

An important ingredient is a beautiful proposition due to Gromov (Proposition 4.4.2) and further work done by Nabutovsky–Rotman [37]. This all together yields

Theorem 1.2.8. If π1(M ) has infinite or finite order ≥ k, then for every pair q, q0 ∈ M

there exist at least k Reeb chords ˜x` from Σq to Σq0 satisfying the time bound

T (˜x`) ≤ kd .

1.2.5 Stability of the minimal time of Reeb chords

Denote by T1(Σ) the minimal time, or the smallest element of the time spectrum of

a given fiberwise starshaped hypersurface Σ. We then consider the C0-stability of the minimal time T1(Σ): Let {Σk}k∈N be a sequence of fiberwise starshaped hypersurfaces

of the cotangent bundle T∗M .

Definition 1.2.9 (C0_{-Convergence of fiberwise starshaped hypersurfaces). The sequence}

(Σk)_k∈N converges in the C0-sense to Σ if for every ε > 0 there exists Nε such that

(1 − ε)D∗Σ ⊂ D∗Σk ⊂ (1 + ε)D∗Σ , for all k ≥ Nε.

We want to understand what happens with the sequence (T1(Σk))_k∈N if (Σk)_k∈N

converges in the C0_{-sense to Σ.}

Proposition 1.2.10. Fix q ∈ M and suppose that {Σk}k∈N converges in the C0-sense

to Σ. Then for all q0 ∈ M it holds true that

T1(Σk) → T1(Σ)

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Chapter 2 Methods and Spaces

This chapter shall explain the basic notions and definitions which will be used later on. We also outline the main methods and theorems that we will apply.

2.1 Background and Setting

Throughout this thesis, let (M, g) be an n-dimensional closed connected Riemannian manifold of diameter d := diam(M ), where n ∈ N is a natural number. Let | · |g be the

norm on the fibers of the tangent bundle T M induced by g. Consider the cotangent bundle T∗M which is isomorphic to the tangent bundle T M via the isomorphism

T : T M → T∗M

(q, v) 7→ (q, αq), where αq(w) = gq(v, w).

Using T , define a Riemannian metric g∗ on the fibers of the cotangent bundle T∗M by

g_q∗(α, β) := gq T−1(q, α), T−1(q, β) , ∀ α, β ∈ Tq∗M, q ∈ M.

Denote the canonical coordinates on T∗M by (q, p).

Definition 2.1.1 (Symplectic manifold, [31]). Let P be C∞-smooth manifold. A sym-plectic structure on P is a non-degenerate closed 2-form ω ∈ Ω2_{(P ), i.e. if}

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2. for p ∈ P and v ∈ TpP it follows that ∀ w ∈ TpP : ωp(v, w) = 0 ⇒ v = 0.

The pair (P, ω) is then called a symplectic manifold.

Note that this definition implies that a symplectic manifold is of even dimension and orientable, see [31].

The cotangent bundle T∗M with the standard Liouville 1-form λ = pdq is the basic example of a symplectic manifold (T∗M, dλ). Choose β > 0 and introduce the Rieman-nian HamiltoRieman-nian function G : T∗_{M → R,}

G(q, p) := βg_q∗(p, p), (2.1)

which will be denoted by G(q, p) =: β|p|2

g∗ =: β|p|2. Let q, q0 ∈ M , and consider the

space of paths γ : [0, 1] → M of Sobolev class-(1, 2) from q to q0,

Ω1_q,q0M =γ ∈ W1,2([0, 1], M ) | γ(0) = q, γ(1) = q0 , (2.2)

as well as the space of continuous paths from q to q0

Ωq,q0M := {γ ∈ C([0, 1], M ) | γ(0) = q , γ(1) = q0} .

Recall the definition of the energy functional E : Ω1

q,q0M → R, E(γ) := 1 2 Z 1 0 | ˙γ(t)|2 gdt (2.3)

and of the length L : Ω1_q,q0M → R of such a path,

L(γ) := Z 1

0

| ˙γ(t)|gdt. (2.4)

For a > 0 we consider the sublevel sets

Ea_{(q, q}0

) := γ ∈ Ω1_q,q0M | E (γ) ≤ a

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as well as

Ω1,a_q,q0M := γ ∈ Ω1_q,q0M | L(γ) ≤ a .

2.1.1 The action functional and the time

A Hamiltonian function is a smooth function on a smooth manifold. Choose a Hamil-tonian function H : T∗_{M → R on the symplectic manifold (T}∗M, dλ). The functional AH: Ω1q,q0T∗M → R, AH(γ) := Z γ∗λ − Z 1 0 H(γ(t)) dt (2.6)

is called the action, and the reduced action or the time is defined by

T (γ) := A(γ) := Z

γ∗λ . (2.7)

We look for paths γ : [0, 1] → T∗M with γ(0) ∈ T_q∗M and γ(1) ∈ T_q∗0M solving

Hamilton’s equations

˙γ = J ∇H(γ(t)) = XH(γ(t)), XH := J ∇H, (2.8)

and denote by P(H, q, q0) the set of all such solutions. The vector field XH : T∗M →

T (T∗M ) is called the Hamiltonian vector field of H. This vector field has a flow called the Hamiltonian flow ϕt_H. The action functional (2.6) is C∞-smooth, and its critical points are precisely the elements of the space P(H, q, q0) of C∞-smooth paths γ : [0, 1] → T∗M solving (2.8). If we specify an action bound AH(γ) ≤ C ∈ R, we denote the set of

solutions which satisfy this action bound by PC_{(H, q, q}0_{) ⊂ P(H, q, q}0_).

Convention 2.1.2. Throughout this thesis we will consider a proper Hamiltonian func-tion K : T∗_{M → R; then its Hamiltonian flow ϕ}t

K exists for all times.

2.1.2 The spherization and fiberwise starshaped hypersurfaces

We are interested in finding solutions of the Hamiltonian equations (2.8). These solutions are Hamiltonian flow lines lying in certain hypersurfaces of the cotangent bundle. Let us describe these so-called fiberwise starshaped hypersurfaces.

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The positive real numbers c ∈ R+ freely act on the cotangent bundle T∗M by

νc : T∗M → T∗M , (q, p) 7→ (q, cp). On T∗M there is the Liouville 1-form λ = pdq and

we have ν_c∗(λ) = cλ, so λ does not descend to the quotient S∗M := T∗_M/R+, but the

kernel does since ker(cλ) = ker(λ) =: ξ. The contact manifold (S∗M, ξ) is called the spherization of the cotangent bundle T∗M . The choice of a nowhere vanishing 1-form α on S∗M with ker(α) = ξ (called the contact form) defines a vector field Rα, called the

Reeb vector field of α, by the two conditions

dα(Rα, ·) = 0, α(Rα) = 1.

Its flow ϕt

α is called the Reeb flow of α. A Reeb chord is a flow line of ϕtα.

To give a more concrete description of the manifold (S∗M, ξ) and the flows ϕt_α, consider a fiberwise starshaped hypersurface Σ of T∗M . We think of it as a smooth hypersurface which is fiberwise starshaped with respect to the zero-section: For every q ∈ M the set Σq = Σ ∩ Tq∗M bounds a set Dq in T∗M , i.e. ∂Dq = Σq, that is strictly

starshaped with respect to the origin 0q ∈ Tq∗M . The restriction λ|Σ of the Liouville

1-form on T∗M to Σ is a contact form for the contact structure ξΣ = ker( λ|_Σ) on Σ,

that gives it the structure of a contact manifold. The diffeomorphism Ψ : Σ → S∗M , (q, p) 7→ q,_kpkp _{· R}+

obtained by radial projection is a contactomorphism, so (Σ, ξΣ)

and (S∗M, ξ) are contactomorphic. One can show that there is a bijection from the set of Reeb flows on (S∗M, ξ) to the set of Reeb flows ϕt

Σ on the set of fiberwise starshaped

hypersurfaces Σ in T∗M . This equivalence gives two ways to study the counting function CFq,q0_,Σ introduced in the Section 2.1.3 from a dynamical point of view.

On the other hand, one can describe our problem in a more geometrical way. We follow Hofer and Zehnder [25, Chapter 4]. Let Σ ⊂ T∗M be a submanifold of the cotangent bundle of codimension one. The cotangent bundle T∗M together with the standard symplectic structure ω := dλ is a symplectic manifold. If we restrict ω to vector fields in T Σ ⊂ T (T∗M ), i.e. if we restrict the 2-form ω to the odd-dimensional subspaces TxΣ ⊂ Tx(T∗M ) for x ∈ Σ, then ω is necessarily degenerate. The kernel of

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this restriction is therefore of dimension one. This defines a line bundle, LΣ ⊂ T Σ,

LΣ = {(x, ξ) ∈ TxΣ | ωx(ξ, η) = 0, ∀ η ∈ TxΣ} .

The line bundle LΣ is called the characteristic line bundle of the hypersurface Σ. This

line bundle gives the direction of every Hamiltonian vector field XH having Σ as a

regular energy surface, i.e. if H : T∗_{M → R is constant on Σ and dH 6= 0 on Σ then} XH(x) ∈ LΣ(x) for x ∈ Σ. Note that LΣ is determined by the hypersurface Σ and by

the symplectic structure ω, hence by geometric quantities. Note that if Σ is interpreted as a contact manifold as above it holds true that the associated Reeb vector field R lies also in LΣ and satisfies trivially λ|_Σ(R) = 1. A characteristic of Σ, or a solution

of ˙x(t) = XH(x(t)), going from one point to another one on Σ, is an embedded open

interval I ⊂ Σ satisfying

T I ⊂ LΣ|I.

The set of characteristics of Σ agrees with the set of unparameterized solutions solving Hamilton’s equations for every Hamiltonian vector field XH on Σ having Σ as a regular

energy surface. So these characteristics agree also with the traces of Reeb chords on the hypersurface Σ.

For more details and examples of fiberwise starshaped hypersurfaces, we refer to Section 2.3.

2.1.3 The counting function CF

q,q0_,Σ

Consider the length spectrum of a Riemannian manifold (M, g) given by

σq,q0(g) = {lengths of all geodesic segments from q to q0}.

In this definition the length is induced by the Riemannian metric g. The study of the length spectrum of a Riemannian manifold is an important problem, with many results and many open questions, see Berger [11, Chapter 10] and Paternain [42, Chapter 5].

Here we are interested in a more general problem: We study the spectrum of a fiber-wise starshaped hypersurface Σ ⊂ T∗M of the cotangent bundle T∗M by interpreting

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it as a contact manifold. Consider the set (the time spectrum of Σ, see Section 2.2.2)

S (Σ, q, q0) := {T (γ) | γ a path on Σ from Σq to Σq0 solving ˙γ = R(γ)} ,

where the number T (γ) is the time (or the reduced action of γ), see (2.7), needed by the Reeb chord γ to go from Σq to Σq0 and R is the Reeb vector field of the contact

manifold (Σ, λ|_Σ). Consider the counting function

CFq,q0_,Σ(T ) = # {τ ∈ S (Σ, q, q0) | τ ≤ T } . (2.9)

If one fixes a time T , then the counting function gives the number CFq,q0_,Σ T of Reeb

chords starting in the fiber Σq and ending in some point of Σq0 before or with the timeT .

Notation (Growth type). Given functions f, g : [0, ∞) → [0, ∞) ∪ {+∞}, we write f < g if there exist constants C, c such that f (t) ≥ g(Ct) + c for all t ≥ 0. Moreover, we write f ≈ g if f _{< g and g < f .}

We say that f has linear growth if f (t) ≈ t, that f has polynomial growth if p _{< f} for some polynomial p, and that f has exponential growth if f (t) ≈ et. We say that f and g have the same growth type if f ≈ g. _♦

In the subsequent chapters we will study the growth type of the counting func-tion CFq,q0_,Σ for different choices of base manifolds M (of T∗M ). This is done by

deriv-ing lower bounds for CFq,q0_,Σ from homological visible geodesic segments of (M, g), see

Definition 4.1.18.

2.2 Lagrangian Floer Homology and the Sandwiching

method

In this section we give a short summary of those parts of Lagrangian Floer homology used later on. We follow Macarini–Schlenk [30] and modify the ideas slightly to get a suitable formulation for our purposes.

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2.2.1 The Hamiltonians G

−

≤ K ≤ G

+

Let Σ ⊂ T∗M be a fiberwise starshaped hypersurface. This property allows one to define a function F : T∗_{M → R by the two requirements}

F |Σ ≡

1

2, F (q, sp) = s

2_{F (q, p)}

for all s ≥ 0 and (q, p) ∈ T∗M .

This function is fiberwise homogenous of degree 2. (If Σ was not fiberwise starshaped, homogeneity of F would not make sense.) Further, F is of class C1 _{and moreover}

smooth off the zero section. To make it smooth, we introduce another smooth function f : R → R, where ε shall be fixed appropriately later on. See Figure 2.1.

                   f (r) = 0, r ≤ ε2 f (r) = r, r ≥ ε f0(r) > 0, r > ε2 0 ≤ f0(r) ≤ 2, _{∀ r ∈ R .}

Then f ◦ F : T∗_{M → R is smooth. Let G : T}∗_{M → R be the usual Riemannian}

ε2

ε f(r)

r

Figure 2.1: The “cut off” function f .

Hamiltonian G(q, p) = 1₂|p|2_{. Multiply the Riemannian metric g of M by a positive}

constant, so the value G(q, p) ∈ R gets scaled independently of (q, p). Hence, we can assume F (q, p) ≥ G(q, p), ∀ (q, p) ∈ T∗M , where the inequality shall be sharp, i.e. there exists a point (q, p) ∈ T∗M such that F (q, p) = G(q, p). We abbreviate this by F ≥ G.

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Then we can choose a positive constant σg ≥ 1 such that σgG ≥ F . The constant

σg = σg(Σ) is called the module of starshapedness of Σ. We refer to Section 2.2.4 for

2.2.2 Hamiltonian action spectra

The action spectrum S (H, q, q0) of a (proper) Hamiltonian function H : T∗_{M → R is} the set of critical values of AH: Ω1q,q0T∗M → R,

S (H, q, q0_{) := {A}

H(x) | x ∈ P(H, q, q0)} .

For b ∈ R define the subsets Sb_{(H, q, q}0_{) := S (H, q, q}0_{) ∩ (−∞, b].}

Let again F : T∗_{M → R be the function with F}−1 1₂ = Σ that is fiberwise homo-geneous of degree 2, and denote by π : T∗M → M the projection along the fibers. We denote by R(Σ, q, q0) the set of Reeb chords from Σq to Σq0. Now we define the time

spectrum of Σ:

S (Σ, q, q0) := {A(γ) | γ ∈ R(Σ, q, q0)} .

Similar to the action spectrum, we set Sb(Σ, q, q0) := S (Σ, q, q0) ∩ (−∞, b]. Lemma 2.2.2. Fix γ ∈ P(F, q, q0).

(i) AF(γ) = F (γ).

(ii) The time of the unique Reeb chord ˜γ ∈ R(Σ, q, q0) for which the trace of π ◦ ˜γ equals the trace of π ◦ γ is A(˜γ) =p2 F (γ).

(iii) In particular, S

√

2b_{(Σ, q, q}0_{) = S}b_{(F, q, q}0_{) for every b > 0.}

Proof. For point (i), see the proof of Lemma 3.1. of [30] with h : x 7→ x. Concerning point (ii), Proposition 4.8.4 implies

γt(x) = ˜γσ(t,x)(x),

where the function σ(t, x) is of the form σ(t, x) = s(x) t with s(x) > 0 constant in the t variable, see the proof of Proposition 4.8.4. Let us calculate,

A(˜γ) = Z 1 0 λ ˙˜γ(t) dt = 1 s(x) Z 1 0 λ ˙˜γ(s(x) t) dt

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= 1 s(x).

The second step is a change of variables and the third step follows because ˜γ is a Reeb chord. On the other hand, we know that F is fiberwise homogeneous of degree two, and in addition that F |_Σ = 1₂. Calculate,

1

2 = F (˜γ(t)) = F ((q(t), s(x)p(t))) = s(x)

2_{F (γ) ,}

what implies that s(x) = √ 1

2 F (γ), therefore A(˜γ) =p2 F (γ) .

The last point (iii) is a direct consequence of point (ii).

Now fix a < b (where a ≤ 0 is not excluded). We can choose ε > 0 in the definition of the function f so small that for every non-constant γ ∈ Pb(f ◦ F, q, q0) we have (f ◦ F )(γ) ≥ ε, and for every non-constant γ ∈ Pb_{(F, q, q}0_{) we have F (γ) ≥ ε. Since}

f (r) = r for r ≥ ε, we then have

S(a,b]_{(f ◦ F, q, q}0

) = S(a,b](F, q, q0) . (2.13)

Furthermore, Proposition 3.2 in [30] shows that γ ∈ Sb_{(K, q, q}0_{) if and only if γ ⊂ {F ≤}

b}. Since K = f ◦ F on {F ≤ b}, we conclude with Lemma 2.2.2 (iii) and (2.13) that Lemma 2.2.3. S(√2a,√2b]_{(Σ, q, q}0_{) = S}(a,b]_{(F, q, q}0_{) = S}(a,b]_{(K, q, q}0₎

for all a, b ∈ R. This lemma generalizes to our Hamiltonians F and K the well-known fact that a geodesic path of length ` has energy 1₂`2_{, see Lemma C.3.4.}

2.2.3 From the Homology of Ω

1_q,q0

M to the time spectrum of Σ

Let Σ ⊂ T∗M be a fiberwise starshaped hypersurface, and let K : T∗_{M → R be the} func-tion constructed in Secfunc-tion 2.2.1. Denote by Ω1

q,q0_,αM the set of W1,2-paths q : [0, 1] → M

with q(0) = q and q(1) = q0that lie in the homotopy class α. We often drop q, q0 from the notation. Given a Riemannian metric g on M , the energy functional Eg: Ω1q,q0_,αM → R

is defined by Eg(q) = 1 2 Z 1 0 g( ˙q(t), ˙q(t)) dt.

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For a < b consider the subsets

Ea_{(q, q}0

, α) =q ∈ Ω1_q,q0_,αM | E_g(q) ≤ a

and the quotient space

QE(a,b](q, q0, α) = Eb(q, q0, α) /Ea(q, q0, α) .

g be a Riemannian metric on M such that q, q0 are not g-conjugate. Scale g such that G ≤ F . Let a < b and σ ≥ σg ≥ 1 be such that

a, b, a/σ, σb /∈ S (G, q, q0) and a/σ, b /∈ S (F, q, q0) .

Then for any field F the number of Reeb chords from Σq to Σq0 in class α with action

in (p2a/σ,√2b] is bounded from below by the rank of the homomorphism induced by inclusion

e

H∗ QE(a/σ,b](q, q0, α) ; F → He∗ QE(a,σb](q, q0, α) ; F .

By eH∗ we denote reduced homology.

Remark. Given a, b, σ, g we find a0, b0, σ0, g0 as close to a, b, σ, g as we like and such that a0, b0, σ0, g0 meet the hypothesis of the proposition. Indeed, the complement of Sb_{(G, q, q}0_{) ∪ S}b_{(F, q, q}0

) in R is open and dense. ♦

Proof of Proposition 2.2.4. We throughout fix q, q0 ∈ M , a, b, σ and g as in the propo-sition, and also fix the field F. The proof is based on Floer homology for Lagrangian intersections. We only recall those properties of Lagrangian Floer homology that we use in the proof, and refer to Section 4 of [30] and the references therein for more details.

Let K : T∗_{M → R be the function constructed in Section 2.2.1. (It depends on} a, b, σg, g.) The Floer chain group CFb(K, α) is the F-vector space freely generated

by the chords in Pb_{(K, q, q}0_{, α). The Conley–Zehnder index (see Section 6.2) of these}

chords (normalized such that it agrees with the Morse index in case of a non-degenerate geodesic chord) gives this vector space a grading ∗. The Floer boundary operator on

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CFb_∗(K, α) is a map of degree −1. Its homology is the Floer homology HFb_∗(K, α). Since the boundary operator maps CFb_∗(K, α) to itself, it descends to the quotient groups

CF(a,b]_∗ (K, α) = CFb_∗(K, α)/ CFa_∗(K, α). The resulting homology is denoted HF(a,b]_∗ (K, α).

The Floer homology of the functions G−, G+ is defined in the same way. There is a

commutative diagram HFb_∗(G−, α) Φ_G−K Φ_G−G+ //_HFb/σ ∗ (G+, α) ASM _// H∗ Eb(α) HFb_∗(K, α) ΦKG+ //HFb_∗(G+, α) ASM _// H∗ Eσb(α) . (2.14)

Here, the three maps Φ between Floer homologies are Floer continuation maps, and ΦG−G+ is an isomorphism. The upper map ASM is the composition

HFb/σ_∗ (G+, α)

AS _//

HMb/σ_∗ (L, α) AM //H∗ Eb(α)

of the Abbondandolo–Schwarz isomorphism from Floer homology to the Morse homology [3] of the Legendre transform L of G+with the Abbondandolo–Mayer isomorphism from

this homology to the homology of Ω1,b

α M [2]. Finally, the two unlabeled vertical arrows

are induced by inclusion.

For the left part of this diagram it is important that the boundaries of the action windows do not belong to the spectrum of the Hamiltonian functions. This is guar-anteed by our assumptions: The definition (2.10) of G− implies that AG−(γ) > b if

γ ∈ P(G−, q, q0) lies outside {G ≤ b}. Since G− = f ◦ G on {G ≤ b}, we thus have

Sb_(G

−, q, q0) = Sb(G, q, q0) and hence S (G−, q, q0). Moreover, b /∈ S (F, q, q0) by

as-sumption, whence b /∈ S (K, q, q0_{) by Lemma 2.2.3. Finally, b/σ, b /}_{∈ S (G}

+, q, q0) because

b, σb /∈ S (G, q, q0_{) and S (G}

+, q, q0) = S (σG, q, q0) = 1_σS (G, q, q0).

The above diagram holds true with b replaced by any c ≤ b, provided that again the boundaries of the action intervals do not belong to the spectrum of the Hamiltonian functions. This is clear for c ≤ 0, and it holds for c ∈ (0, b] in view of the computation in the proof of Lemma 3.3 of [30], provided we choose ε = ε(c) in the the definition of f small enough. In particular, our assumptions imply that the diagram holds with b replaced by a/σ. Since the homomorphisms in the above diagram are all defined at the

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chain level, we then obtain the commutative diagram HF(a/σ,b]_∗ (G−, α) ∼ = _// HF(a/σ_∗ 2,b/σ](G+, α) ∼ = _// e H∗ QE(a/σ,b](α)

HF(a/σ,b]∗ (K, α) //HF(a/σ,b]∗ (G+, α) //He∗ QE(a,σb](α) .

(2.15)

By eH∗ we denote reduced homology. It follows that the cardinality of P(a/σ,b](K, α) is at

least the rank of the right vertical map. The theorem follows together with Lemma 2.2.3.

For later reference, we state the “absolute case” separately:

g be a Riemannian metric on M such that q, q0 are not g-conjugate. Scale g such that G ≤ F . Let b > 0 be such that

b /∈ S (G, q, q0) ∪ S (F, q, q0) .

Then for any field F the number of Reeb chords from Σq to Σq0 in class α with action in

[0,√2b] is bounded from below by the rank of the homomorphism induced by inclusion

H∗ Eb(q, q0, α) ; F → H∗ Eσb(q, q0, α) ; F .

Remark. In Proposition 2.2.4 we assumed in addition, this in contrast to the situation in Proposition 2.2.5, that b/σ /∈ S (G, q, q0_). _{Note that this condition is implicitly}

satisfied as it follows directly from Macarini–Schlenk [30, Proposition 3.3]. _♦

2.2.4 The module of starshapedness σ

g

In Section 2.2.1 the module of starshapedness of Σ was introduced. We give a geometric interpretation. By assumption it holds true that Σ = F−1 1₂. Comparing this set to G−1 1₂ and to (σgG)−1 1₂ shows that these hypersurfaces are nested or “sandwiched”:

First, Σ lies – by touching its boundary at least at one point – in the bounded part of G−1 1₂, while (σgG)−1 1₂ is enclosed by Σ. If Sr∗M = {(q, p) ∈ T

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r-co-sphere bundle, we can see G−1 1₂ = {(q, p) ∈ T∗M | (q, p0) ∈ S₁∗M and p = routerp0},

so G−1 1₂ = S_r∗_outerM , and similarly we get (σgG)−1 1₂ = Sr∗innerM for the minimal

router > 0 and the maximal rinner > 0. So, for x = (q, routerp0) ∈ Sr∗outerM we have

G(x) = 1₂r2 outer = 1 2 router rinner rinner p 0 2 = router rinner 2 1 2|rinnerp 0_|2 = σgG(·, rinnerp0). Therefore, σg = router rinner 2

. If a result does not depend on the actual choice of the Riemannian metric g, one can choose σg to be independent of g by considering the following (smaller)

constant of starshapedness: σ := infg

router rinner

2

. See Figure 2.3 for an illustration.

T∗ qM Σq (σgG) −1 1 2 q G−1 1 2 q 0q

Figure 2.3: The sandwiching of Σ by the two co-sphere bundles (σgG)−1 1₂ and G−1 1₂

restricted to the fiber T_q∗M . Note that Σqmust intersect (σgG)−1 1₂

qand independently

G−1 1₂

q for at least one q ∈ M .

We next give a class of examples of fiberwise starshaped hypersurfaces for which one can calculate the module of starshapedness.

Examples. (Physical Hamiltonians). Consider a physical Hamiltonian

Hphys: T∗M −→ R,

(q, p) 7−→ Hphys(q, p) := β|p|2+ V (q) .

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and a smooth function V : M → R. If V (q) < 12, then Σ := H −1 phys

1

2 is fiberwise convex

w.r.t. 0q ∈ Tq∗M . Since M is a closed manifold, V attains its maximum and minimum.

Denote them by Vmax := maxx∈MV (x) < 1₂ and Vmin := minx∈MV (x). For the maximal

and the minimal momenta we compute

|p|2 max = 1 2 − Vmin β , |p| 2 min= 1 2 − Vmax β . Hence, σg = router rinner 2 = |p|max |p|min 2 = 1 2 − Vmin 1 2 − Vmax .

(Co-sphere bundles). If the fiberwise starshaped hypersurface Σ ⊂ T∗M is a co-sphere bundle, i.e. if it is given by Σ = {(q, p) ∈ T∗M | gq(p, p) = r} for some r ∈ R,

then σg(Σ) = 1. ♦

2.3 Fiberwise Starshaped Hypersurfaces

This section is devoted to examples of fiberwise starshaped hypersurfaces in the cotan-gent bundle of a closed connected and finite-dimensional smooth manifold M .

Examples. We give three classes of fiberwise starshaped hypersurfaces on some base manifolds M .

(Co-sphere bundles). Let (M, g) be a closed connected Riemannian manifold. If the fiberwise starshaped hypersurface Σ ⊂ T∗M is a co-sphere bundle, i.e. if it is given by Σ = {(q, p) ∈ T∗M | gq(p, p) = r} for some r ∈ R, then Σq = Σ ∩ Tq∗M is a circle of

radius√r in T_q∗M for every q.

(Level sets of physical Hamiltonians). Let Hphys be a physical Hamiltonian

Hphys: T∗M −→ R,

(q, p) 7−→ Hphys(q, p) := β|p|2+ V (q) .

as it introduced in Section 2.2.4. Consider a regular level set Σ of Hphys. The set

Σq = Σ ∩ Tq∗M is as before a circle in every fiber, but to the contrary, the radius of the

Counting Reeb Chords on spherizations

Counting Reeb Chords on

Spherizations

THÈSE DE DOCTORAT

Raphael Elias Wullschleger

IMPRIMATUR POUR THESE DE DOCTORAT

La Faculté des sciences de l'Université de Neuchâtel

autorise l'impression de la présente thèse soutenue par

Monsieur Raphael WULLSCHLEGER

Titre:

“Counting Reeb Chords on Spherizations”

Summary

Acknowledgements

Meinen Eltern gewidmet

Verena Wullschleger–Basler

und

Table of Contents

Notations and Symbols

Chapter 1

Introduction

1.1

The questions of this thesis

1.2

Summary of the results

1.2.1

Exponential and polynomial growth of the number of

so-lutions

1.2.2

Time bounds for the first and the second Reeb chord

1.2.3

Time bounds for the first k Reeb chords – via Morse theory

1.2.4

Time bounds for the first k Reeb chords – via topology

1.2.5

Stability of the minimal time of Reeb chords

Chapter 2

Methods and Spaces

2.1

Background and Setting

2.1.1

The action functional and the time

2.1.2

The spherization and fiberwise starshaped hypersurfaces

2.1.3

The counting function CF

2.2

Lagrangian Floer Homology and the Sandwiching

method

2.2.1

The Hamiltonians G

≤ K ≤ G

2.2.2

Hamiltonian action spectra

2.2.3

From the Homology of Ω

M to the time spectrum of Σ

2.2.4

The module of starshapedness σ

2.3

Fiberwise Starshaped Hypersurfaces